University of Tehran Journal of Sciences, Islamic Republic of Iran 1016-1104 22 2 2011 06 01 Generalized Baer-Invariant of a Pair of Groups and the Direct Limit Generalized Baer-Invariant of a Pair of Groups and the Direct Limit 159 164 23040 EN M.R. Rismanchian Department of Mathematics, Faculty of Sciences, University of Shahre-Kord Journal Article 1970 01 01 In this paper we introduce the concept of generalized Baer-invariant of a pair of groups with respect to two varieties ? and ? of groups. We give some inequalities for the generalized Baer-invariant of a pair of finite groups, when ? is considered to be the Schur-Baer variety. Further, we present a sufficient condition under which the order of the generalized Baer-invariant of a pair of finite groups divides the order of the generalized Baer-invariant of a pair of their factor groups. Moreover, we prove that the generalized Baer-invariant of a pair of groups commutes with direct limit. https://jsciences.ut.ac.ir/article_23040_8d115e62f8f1921f064a4d30749a5c09.pdf